On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one

TL;DR

This paper extends results on Fermat's equation over quadratic imaginary fields of class number one, showing non-existence of certain solutions for large primes, based on a generalized Serre's modularity conjecture.

Contribution

It generalizes previous work by proving non-existence of specific solutions to Fermat's equation over these fields under a conjectural modularity assumption.

Findings

No solutions for primes p ≥ 19 with certain divisibility conditions

Results depend on a conjectural generalization of Serre's modularity conjecture

Extends prior results to broader class of quadratic imaginary fields

Abstract

In the present article, we extend previous results of the author and we show that when $K$ is any quadratic imaginary field of class number one, Fermat's equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K \setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.